Remarks on Existence of Noise-Induced Long-Time Tail

Existence of noise-induced long-time tail (critical slowing down) is studied. Exact analytic solution of the stochastic differential equation dXt=(γXt-Xtn)dtγXt°dWt, where Wt is the Wiener process, is confirmed by solving the hierarchy equations of moments ≪Xtp>. The simple deriving procedure gives the same results as those obtained by Suzuki et al. and Brenig and Banai. Transition probability density function corresponding to the stochastic differential equation is also derived. The moments ≪Xtp> are alternatively described with the help of the transition probability density function. It is revealed, by numerical integrals, that the first few moments behave as t-1/2 for large t. Namely the noise-induced long-time tail occurs in the present model near the critical point. Monte Carlo simulations of the stochastic differential equation also confirm the occurrence of the noise-induced long-time tail. The numerical calculations are made in the case n=3.